The Tidal Response of Rocky and Ice-Rich Exoplanets G

Astronomy & Astrophysics manuscript no. Tobie-AA_Tides_exoplanets c ESO 2018 August 30, 2018

The tidal response of rocky and ice-rich exoplanets G. Tobie1, O. Grasset1, C. Dumoulin1, and A. Mocquet1

Laboratoire de Planétologie et Géodynamique, UMR-CNRS 6112, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France e-mail: [email protected]

Submitted Sept. xx, 2018

ABSTRACT

The amount of detected planets with size comparable to the Earth increases drastically. Most of the Earth-size planet candidates orbit at close distances from their central star, and therefore are subjected to large tidal forcing. Accurate determination of the tidal parameters of exoplanets taking into account their interior structure and rheology is essential to better constrain their rotational and orbital history. In the present study, we compute the tidal response of rocky and ice-rich solid exoplanets for masses ranging between 0.5 and 10 Earth masses using a multilayer approach and an Andrade rheology. We showed that the amplitude of tidal response, characterized by the gravitational Love number, k2, depends mostly on the planet mass and the internal stratification (mostly controlled by the bulk composition). We showed that, for a given planet composition, the tidal Love number, k2, is mostly controlled by self-gravity effect and increase as a function of planet mass. For rocky planets, k2 depends mostly on the relative size of the iron core, and hence on the bulk iron fraction. For ice-rich planets, the presence of outer ice layers reduces the amplitude of tidal response compared to ice-free rocky planets. For both types of planet (rocky and ice-rich), we propose relatively simple scaling laws to predict the Love number as a function of radius, planet mass and composition. For the dissipation rate, characterized by the Q factor, we did not find any direct control by the planet mass. The dissipation rate is mostly sensitive to the forcing frequency and the internal viscosity, which depends on the planet composition and size.

Key words. Exoplanets – Interiors – Tides

1. Introduction in the TRAPPIST-1 system, even if the orbital eccentricity is estimated to be small, strong internal heating, exceeding The number of detected exoplanets with radius and mass com- radiogenic heating, can be generated for the closest planets (e.g. parable to the Earth is now increasing drastically (e.g. Bonfils Turbet et al. 2018). The impact of tidal interactions on planetary et al. 2013; Batalha et al. 2013; Marcy et al. 2014; Dressing & evolution depend obviously on the orbital configuration of the Charbonneau 2015; Coughlin et al. 2016; Dittmann et al. 2017; system, but also on the way planetary bodies deform under the Gillon et al. 2017). Most of the low-mass planet candidates action of tides raised by the central stars, and in some cases by orbit at relatively close distance from their central star and planetary companion, like in the Earth-moon system. therefore are likely subjected to large tidal forcing. Low mass planets (2 − 10ME) with short orbital periods (< 10 − 20 days) seem especially abundant around M-dwarf stars (e.g. Bonfils The response of planetary interiors to tidal forcing is et al. 2013; Dressing & Charbonneau 2015). For example, the dependent on the internal structure and on the mechanical TRAPPIST-1 system recently discovered by Gillon et al. (2016, properties of each layer composing the planetary interior. Many 2017) exhibits several small planets orbiting a low-mass star at models have been developed to compute the tidal response of relatively close distance (< 0.1 AU), corresponding to orbital a variety of planetary objects of the Solar system in the past periods of a few Earth days. using multi-layer methods (e.g. Alterman et al. 1959; Kaula 1964; Sabadini et al. 1982; Segatz et al. 1988; Tobie et al. 2005; Such multi-planet system evolves mainly due to the grav- Wahr et al. 2009; Beuthe 2013; Dumoulin et al. 2017). Most itational tide raised by the star in the planets (the planetary of studies dedicated to exoplanets used simplified approach to tide). The planetary tide mainly acts to decrease the obliquity predict the tidal response assuming, for instance, the formula of the planet, synchronize the rotation and on longer timescales derived for homogenous viscoelastic interiors (Henning et al. decrease the eccentricity and semi-major axis. Therefore, tidal 2009; Efroimsky 2012; Makarov & Efroimsky 2014; Barr et al. evolution has a strong impact on the climate stability of the plan- 2017), thus neglecting the effect of density stratification and the ets and their habitability (e.g. Lammer et al. 2009; Barnes 2017; mechanical coupling between the different internal layers. To Turbet et al. 2018). Tidal friction occurring in the interior of our knowledge, only the study of Henning & Hurford (2014) such planets during tidal despinning as well as once the planet is used a multi-layer method to predict the tidal response of a tidally locked on an eccentric orbit can, in some circumstances, variety of exoplanets. However, Henning & Hurford (2014) significantly contribute to the internal heat budget, possibly considered interior models with constant values of density, enhancing internal melting and volcanism (Behounkováˇ et al. rigidity and viscosity for each internal layer 2011; Henning & Hurford 2014; Barr et al. 2017). For instance

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With the growing number of detected exoplanets, a variety magma ocean or liquid water ocean). This particular case will of structural models have been developed this last decade to be addressed in a future study. The input parameter are Fe/Si, predict the internal structure of massive rocky planets and large Mg/Si, the Mg content of the silicate mantle, the amount of icy worlds (e.g. Valencia et al. 2006; Sotin et al. 2007; Seager H2O and the total mass of the planet. For simplicity, we impose et al. 2007; Grasset et al. 2009; Swift et al. 2012; Weiss & a Mg content in the silicate mantle equal to the Earth’s one Marcy 2014; Dorn et al. 2015). These studies showed that the (defined as the mole fraction Mg/(Mg+Fe): Mg#=0.9) for mass-radius relationship is sensitive to the planet composition, the two classes of interior models. The reference value for so that accurate determination of planet mass and radius com- Fe/Si and Mg/Si are fixed to the solar value, 0.977 and 1.072 bined with stellar elemental abundances may potentially provide respectively. For the rock planets, the Fe/Si ratio is varying constraints on the core size, mantle composition and the water between δFe = −50% and +50% relative to the reference value. fraction of the planet. Even if the uncertainties are still relatively For ice-rich planets, the Fe/Si ratio is fixed to the reference high on the mass determination of small-size planets (e.g. Weiss value and only the ice fraction (xice is varied between 0 and 50% . & Marcy 2014), we can already observe some compositional tendencies. Rogers (2015) noticed, for instance, that most of The interior structure is modeled using the approach of Sotin planets with radius larger than 1.6 R⊕ have density lower than et al. (2007). The same equations of state (EoS) and parameters terrestrial planets suggesting a significant fraction of volatile, in are used for all layers, except for the surface liquid water layer form of H/He atmosphere and/or water layers. which replaces by low-pressure ices. The density profile in the upper part of the silicate mantle (P < 25 GPa) as well as in In models studying tidal evolution of planetary rotations and the low-pressure ice layers (P < 2.2 GPa) is computed with orbits (e.g. Mardling 2007; Bolmont et al. 2014; Barnes 2017), a Birch-Murnaghan EoS. For simplicity, no phase transition the effect of tidal damping is classically parametrized using is considered in the upper rock mantle (P<25 GPa) and the prescribed values of tidal Love number, k2, and dissipation low-pressure ice layers. These two low-pressure layers are function, Q−1 or time lag ∆t. In reality, the tidal response described with a single set of parameters. For the liquid iron is expected to depend on the planet composition, structure, core, the lower part of the silicate mantle and the high-pressure thermal state as well as on the frequency of the tidal forcing. ice layer (Ice VII), a Mie-Grunëisen-Debye EoS is employed. This requires to take into account the specificity of the interior The iron-rich core is assumed to be entirely liquid, no inner models and orbital configuration to predict the appropriate tidal solid core is considered. In all layers, the temperature profile parameters. is assumed to follow an isentropic temperature profile. Figure 1 displays the Mass-Radius relationship computed for interior In the present study, we perform systematic computation models with various iron and ice contents and the comparison of the tidal response for planet masses ranging between 0.5 with scaling laws using the polynomial coefficients provided in and 10 M⊕, having various Fe/Si ratio and H2O content. For Table 1. that purpose, we use a numerical code initially developed for computing the tidal response of icy moons (Tobie et al. 2005), recently applied to Venus and Earth’s application (Dumoulin et al. 2017). To determine the internal structure of planet as a function of mass and composition, we follow the method of Sotin et al. (2007) (see section 2). The rheological properties are adapted from the Earth’s case assuming an Andrade viscoelastic rheology and are extrapolated to high-pressure ranges (see section 2). The method for computing tidal deformation is δFe=+50% presented in section 3. A validation of the structural and tidal δ = 0% computation is then presented on the Earth’s case in section 4. Fe δFe=-50% Section 5 displays the results for rocky planets, with various x =+10% Fe/Si ratio, while section 6 is devoted to ice-rich planets. For ice x =+20% both types of planets, we show that the tidal Love number can ice x =+50% be described as a function of planet mass using a rather simple ice relationship. We also present the sensitivity of the dissipation function to planet mass, tidal frequency and assumed viscoelastic parameters. Some applications of our derived scaling laws to a selection of exoplanets are finally provided in section ??. Fig. 1. Mass-Radius relationship, normalized to the Earth, obtained from rocky and ice-rich planets with various iron content (−50% ≤ δFe ≤ +50%) and ice fraction (0 ≤ xice ≤ 50%) . Each point corre- 2. Interior structure and rheology sponds to the results obtained using the modeling approach described We consider two classes of planets: (1) rocky planets with above, based on Sotin et al. (2007). The lines correspond to the scaling laws using the coefficients provided in Table 1, determined as a function global iron fraction varying between -50% and +50% relative to of planet composition using a polynomial fit of the modeled structures. the Earth, (2) planets enriched in water ice, with H2O fraction up to 50%. For all planet classes, we consider planet mass ranging between 0.5 and 10 ME. The interior is assumed to The elastic bulk isentropic modulus, K, are derived from the dP be differentiated in a metallic core (assumed fully liquid), a density and pressure profiles: K = ρ dρ . The shear modulus, µ, in silicate mantle, divided in an upper and lower mantle, and, each solid layer is then estimated from the bulk modulus and the for ice-rich planets, a thick ice layer. We do not consider the pressure. For the silicate part, the following relationship, which possible presence of external fluid envelops (dense atmosphere, reproduces well the shear modulus profile in the Earth’s mantle

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Table 1. Polynomial coefficients derived to reproduce the M-R relation- β thermal convection should tend to limit the viscosity variation ship (R/R⊕ = α(M/M⊕) ) as a function of planet composition for rocky 2 2 throughout the layers, so that the approximation of constant and ice-rich planets: α = a0 + a1 x + a2 x ; β = b0 + b1 x + b2 x viscosity remains reasonable. As shown in the case of Venus, differences between solutions with variable viscosity with Rocky Ice-rich depth and solutions with constant viscosity remains small and (x = δ ) (x=x ) Fe ice can be reasonably neglected at first order (Dumoulin et al. 2017). α a 0.99781 0.99786 0 For each layer, we define a reference viscosity which is a -0.05536 0.62422 1 equal to 5.1020 Pa.s in the low-pressure rock mantle layer, 5.1022 a 0.01890 -0.33042 2 Pa.s in the high-pressure rock mantle layer (corresponding to β the upper mantle and lower mantle in the Earth), and 5.1025 Pa.s b 0.272728 0.273249 0 in the very-high-pressure rock mantle, existing for M ≥ 1M⊕. b1 -0.001106 -0.010867 b2 0.00062 -0.019278 3. Computation of the viscoelastic tidal response (Dziewonski & Anderson 1981; Stacey & Davis 2008), is used: The viscoelastic deformation of the planet interior under the µ = 0.631−0.899 P . For the ice layers, a similar relationship K sil K action of periodic tidal forces is computed following the method is used and constrained from existing experimental data at high of Tobie et al. (2005), adapted to terrestrial planets in Dumoulin pressures (Polian & Grimsditch 1983): µ = 0.6 − 0.9 P . et al. (2017). The iron core is supposed to be fully fluid, K ice K inviscid and incompressible and all the other layers consists To determine the viscoelastic properties of the interior, we of viscoelastic compressible solids. The possible presence of use an Andrade model (Castillo-Rogez et al. 2011; Efroimsky external fluid envelop (dense atmosphere, magma ocean or 2012), which allows the calculations of complex modulus from water ocean) is not considered and will be addressed in a future the elastic shear modulus µ, the elastic bulk modulus K and the study. Using the density profile and rheological properties viscosity. The complex compliance, which corresponds to the computed in section 2, the Poisson equation and the equation inverse of the complex shear modulus (µc(χ) = 1/J(χ)), is given of motions are solved for small perturbations in the frequency for the Andrade model by: domain assuming a compressible Andrade viscoelastic rheology (e.g. Castillo-Rogez et al. 2011). 1 i J(χ) = − + β (iχ)−α Γ (1 + α) , (1) µ ηχ The Love number k2, characterizing the potential perturba- tion, the dissipation function, Q−1 are computed by integrating where χ is the tidal frequency, α and β are parameters describing the complex radial functions associated with the radial and the frequency dependence and the amplitude of the transient tangential displacements (y1 and y3, respectively), the radial response, respectively. Comparison with available experimental and tangential stresses (y2 and y4, respectively), and the grav- data for rock and applications to the Earth (see section 4) indi- itational potential (y5), as defined by Takeushi & Saito (1972). cate that the Andrade model is a good approximation to describe The formulation of the spheroidal deformation developed by the anelastic attenuation at tidal frequencies (Castillo-Rogez Takeushi & Saito (1972) was initially derived for the elastic et al. 2011). For the α parameter, we explore a range of values case. Nevertheless, as explained in Tobie et al. (2005), the same varying between 0.2 and 0.3, which frames the typical value formulation can be used for the viscoelastic case by solving required to explain the Q factor of the Earth’s mantle (see it in the frequency domain and by defining complex shear section 4). For the β parameter, following the approximation and bulk modulii, equivalent to the elastic modulii used in the of Castillo-Rogez et al. (2011), we assume that β ' µα−1η−α, elastic problem. For more details, please see Tobie et al. (2005) which is justified for olivine minerals (Tan et al. 2001; Jackson where applications to icy moons using a Maxwell rheology et al. 2002). is presented. Here we use the Andrade rheological model to define the appropriate complex modulus. As shown in , this The viscosity is assumed to increase due to phase transition viscoelastic rheology reproduce well the dissipate function of (ice I > HP ice; Low-pressure (LP) rock mantle > High-pressure the Earth’s interior as well as its frequency dependence over a (HP) rock mantle > Very-High-Pressure (VHP) rock mantle) wide range of frequencies. or due to composition change (ice > rock). For sake of simplicity, the viscosity is assumed constant in each sublayer. The For the fluid core, the simplified formulation of Saito (1974) 14 16 viscosity in the ice layers typically varies between 10 − 10 relying on to radial functions, y5 and y7, is employed assuming Pa.s for low-pressure ice (ice I to Ice VI) and 1016 − 1018, a quasi-static and non-dissipative fluid material. The solution in and possibly even larger values, for high-pressure ices (ice the solid part of the interior is expressed as the linear combi- VII) (e.g. Durham et al. 1997). Based on Earth’s estimates, nation of three independent solutions, which reduces to two in- the viscosity in the silicate mantle typically varies between dependent solutions in the fluid part. The system of six differ- 1019 − 1021 Pa.s in the upper mantle and 1022 − 1023 Pa.s in the ential equations is solved by integrating the three independent lower mantle (e.g. Cížkováˇ et al. 2012, and references therein). solutions using a fifth-order Runge-Kutta method with adjus- For larger planets, the rock viscosity is expected to strongly tive stepsize control from the center (r = 0 km) to the planet 25 26 increase above ∼ 120 GPa, possibly up to 10 − 10 Pa.s surface (r = RP km) and by applying the appropriate boundary (e.g. Karato 2011; Tackley et al. 2013). The effect of pressure, conditions (see for more details: Takeushi & Saito (1972); Saito ∗ especially at elevated pressure, is not well constrained and (1974); Tobie et al. (2005)). The complex Love numbers, k2 are large uncertainties remain on the possible values of viscosity determined from the radial functions, y5(RP) at the planet sur- −1 in the lowest part. However, as shown by Tackley et al. (2013), face (r = RP), and the global dissipation function, Q by the

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∗ ratio between the imaginary part and the module of k2:

∗ −1 ∗ ∗ k2 = y5(RP) − 1; Q = =(k2)/kk2k. (2)

4. Model validation on the Earth case

6000

5000

4000

3000 R (km) 2000 Fig. 3. Computed Q factor of the Earth’s solid mantle using the synthetic 1000 proﬁle displayed in Fig. 2 for various values of viscosity for the lower mantle, ηLM (with ηLM/ηUM = 100) and of α parameter for the Andrade rheological model. The bold black line indicated the estimated value of 2 4 6 8 10 1214 0 1000 2000 0 100 200 300 400 Q’s Earth interior and the grey bound the uncertainty range estimated -3 ρ (x 1000 kg.m ) Ks (GPa) μ (GPa) by Ray et al. (2001). Fig. 2. Comparison between our synthetic proﬁles (solid line) of density, ρ, bulk modulus, K and shear modulus, µ, derived following the method described in section 2 to those of the Preliminary Reference Earth Model (PREM) (Dziewonski & Anderson 1981) (dotted line), as- δFe=+50% suming an interior composition corresponding to the reference case . δFe=+25%

δFe= 0% Figure 2 compares our synthetic profiles of density, ρ, bulk δ =-25% modulus, K and shear modulus, µ, derived following the method Fe described in section 2 and using the reference parameters δFe=-50% (Fe/Si=0.977, Mg/Si=1.072, Mg#=0.9), to the Preliminary Ref- erence Earth Model (PREM) (Dziewonski & Anderson 1981). The density and bulk modulus is well reproduced throughout the entire silicate mantle, with just some slight departure in the upper mantle as we neglect any phase transition occurring in this layer. The shear modulus in the upper mantle is also a little bit larger than the PREM values as, for simplicity, we represent µ as the function of K and P using a single formula (see section 2), which is more representative of the lower mantle. Obviously, as we neglect the presence of a solid inner core, the density is Fig. 4. Computed tidal Love number k2 for rocky planets with mass ranging between 0.5 and 10 Earth’s mass and iron content varying be- underestimated in the inner part of the core, which has only tween -50% and +50% relative to the reference value. The lines repre- minor effects on the tidal response. sent the fit based on Eq. 3 and the polynomial coefficient provided in Table 3 For the viscosity profile, we assume constant values in the upper and lower mantles, ranging between 1020 − 1021 Pa.s and 1022 − 1023 Pa.s, respectively, consistent with the typical ranges (0.2 − 2ηre f ), the Earth’s Q value of about 230-360 in- viscosity values derived from geophysical constraints (see ferred from satellite tracking and altimetry (Ray et al. 2001) can Cížkováˇ et al. (2012) and reference therein). For this range be reproduced for α values between 0.23 and 0.28. For this range of viscosity, α parameters between 0.2 and 0.3 and using the of α, we verify also that we reasonably reproduce the frequency PREM profiles for ρ, µand K, we obtain a Love number for dependence of the Earth’s Q inferred from the fortnightly M f semi-diurnal tidal period between 0.302 and 0.306, which is tide (13.66 days, Ray & Egbert 2012), the Chandler Wooble (433 consistent with the observed value estimated between 0.304 days, Furuya & Chao 1996; Benjamin et al. 2006) and the 18.6- and 0.312 (Ray et al. 2001). At the period of 18.6 yr, we also yr tide (Benjamin et al. 2006), (see Table 2). obtained a k2 value consistent with the value inferred from observations. Using the synthetic profiles, we obtain a slightly 5. Results for rocky planets smaller value (0.299-0.302 at 12.42 hr and 0.312-0.336 at 18.6 yr) due mostly to the overestimate shear modulus in the upper Fig. 4 displays the computed tidal Love number, k2, as a func- mantle. This remains, however, very close to the Earth’s value, tion of planet mass for three iron contents ranging between -50% the difference being less than 1%. and +50%, relative to the reference value. It shows that planets with higher iron content are characterized by higher Love num- As illustrated on Fig. 3, the modeled Q value is very sensi- ber, which is explained by the larger size of the liquid iron core, tive to the assumed values of mantle viscosity and α parameter. resulting in a larger deformation of the silicate mantle. For all At the period of semi-diurnal tide (M2), for assumed viscosity tested values of iron content, we notice that the increase of Love

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Table 2. Comparison between observed and computed k2 and Q at diﬀerent forcing periods for the solid Earth (corrected from ocean tides), using a b c d 0.23 ≤ α ≤ 0.28, 0.2 × ηre f ≤ η ≤ 2 × ηre f . References: Ray et al. (2001); Benjamin et al. (2006); Ray & Egbert (2012); Furuya & Chao (1996)

12.42 hr 13.66 d 433 d 18.6 yr Observed a b k2 0.304 − 0.312 - - 0.337 − 0.340 Q 230 − 360a 65 − 135c 50 − 150d 10 − 30b

Computed k2 (PREM) 0.302 − 0.306 0.304 − 0.311 0.309 − 0.323 0.318 − 0.341 k2 (synth.) 0.299 − 0.302 0.300 − 0.306 0.304 − 0.318 0.312 − 0.336 Q 200 − 380 80 − 190 40 − 80 20 − 40

Table 3. Polynomial coeﬃcients derived to reproduce the Love number as a function of composition and mass planet for rocky and ice-rich re f 2 3 1 day planets: k2 (x) = a0 + a1 × x + a2 × x + a3 × x and δk2(x) = b0 + b1 × 2 3 x + b2 × x + b3 × x 25 ηPP=5.10 Pa.s

Rocky Ice-rich δFe=+50% (x = δFe) (x=xice) re f k2 a 0.29946244 0.29949203 0 22 δFe=-50% ηPP=5.10 Pa.s a1 0.16901399 -1.1512161 a2 -0.021502733 3.3202312 a3 -3.1655305 δk2 b0 0.17098079 0.17302082 b1 -0.0078913109 -0.54128325 b2 0.0070187880 1.7670414 b3 -1.7880138

25 1 day ηPP=5.10 Pa.s δFe=0% number with mass can be scaled to the surface gravity relative to 10 days the Earth using the following relationship: 100 days     re f  2  k2(δFe) = k (δFe) + 1 −  × δk2(δFe) (3) 2  h g i2   1 +  g⊕ with M, R and g the mass, radius and gravity (⊕ stands for 22 the Earth). We then derived the variations of the coefficients, ηPP=5.10 Pa.s re f k2 (δFe) and δk2(δFe), as a function of iron content using a polynomial fit to degree 2 with x = δFe. The coefficients derived from this polynomial fit are provided in Table 3. As shown on Fig. 4, this simple relationship reproduces relatively well the Love number over the whole range of mass tested here (0.5-10 M⊕). The tidal frequency and assumed viscosity have only a Fig. 5. Computed Q factor for rocky planets with mass ranging between very minor effect on k2 as the tidal response is dominated by the 0.5 and 10 Earth’s mass (a) for a tidal period of 1 day and three iron elastic response for the range of tidal period tested here (0.5-100 content values varying between -50% and +50% relative to the refer- days). ence value, and (b) for Earth’s iron content and three tidal periods of 1 (square), 10 (triangle) and 100 (diamond) days. The extreme value are 22 For the dissipation factor, Q, we also observe a systematic considered for the very-high-pressure rock viscosity (ηVHP): 5.10 Pa.s (grey) assuming no viscosity increase relative to the reference Earth increase as a function of planet mass as well as a dependency lower mantle and 5.1025 Pa.s (black) assuming a 1000 factor increase with iron content (which controls the ratio between mantle relative to the reference Earth’s lower mantle. thickness and core radius). However, Q is much more sensitive to tidal frequency and mantle viscosity than the mass and internal structure (Figure 5). As expected (e.g. Efroimsky 2012), mass. it decreases with increasing forcing periods and decreasing viscosities. For high viscosity values in the very-high-pressure We did not find any straightforward scaling for the Q vari- rock layer, the dissipation factor, Q, increases with planet mass ations as a function of planet mass. Nevertheless, for a given as the proportion of the post-perovskite layer in the rock mantle planet mass, we found that the variation of Q as a function of and hence the average viscosity of the mantle increase with tidal frequency, viscosity, iron content and α parameter can be

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1 day xice= 0%

xice= 10%

xice= 20%

xice= 50%

100 days

Fig. 6. Scaling of the Q factor as a function of mantle viscosity η, tidal Fig. 8. Computed Q factor for ice-rich planets with mass ranging be- frequency χ, α Andrade parameter and iron content, δFe, relative to ref- tween 0.5 and 10 Earth’s mass for two tidal periods (1 day in black 22 erence values (ηre f = 5.10 Pa.s, χre f = 2π/1day), for four planet and 100 days in grey) and three ice fraction (10%(cross), 20% (trian- 18 masses (1, 2, 5 and 10 M⊕). gle) and 50% (diamond)) for a HP ice viscosity of 10 Pa.s and the reference viscosity profile for the rocky part (see section 2). reproduced with the following relationship: For ice-rich planets, we observed similar tendencies as a (A+Bα) α α Q = 10 × (η/ηre f ) × (χ/χre f ) , (4) function of mass than for rocky planets (Fig. 7). The dependen- cies of the k2 coefficients (Eq. 3) as a function of ice content can with ηre f being the reference value for the viscosity profile (see be well reproduced using a polynomial fit up to degree 3 with section 2) and χre f = 2π/1day. As illustrated on Figure 6, this x = xice, corresponding to the mass fraction of ice relative to the scaling reproduces very well the Q factor for a wide range of total planet mass. The corresponding coefficients are provided viscosity, frequency and iron content. This scaling remains valid in Table 3. as long as the tidal period is much smaller than the Maxwell time, τM, defined as the ratio between the viscosity, η, and the As shown on Figure 8, Q varies only moderately with planet shear modulus, µ. For the range of viscosity tested here, the mass. It is mostly sensitive to the ice fraction, viscosity and Maxwell time is always larger than 100 years (up to 1 million forcing frequency. Like for the rocky planets, we noticed a years for massive highly viscous exoplanets), which is much systematic dependency with frequency and viscosity. However, longer than the typical tidal period (< 100 days). for ice-rich planets, this dependency is more complex as the Maxwell time of the ice layer, τM = η/µ, is much shorter and hence comparable to the tidal period. For ηχ values ranging between 109 and 1010 Pa, Q reaches a minimal value, which 6. Results for ice-rich planets varies between 2 and 7 depending on the ice content and planet mass (Fig. 9). For ηχ 1010 Pa, Q follows a behavior comparable to the rocky planets with Q ∼ ηαχα. When approaching values of typically 1010 − 1011 Pa, the frequency-viscosity = 0% xice dependency changes to Q ∼ ηχ. Then, after the minimal Q value is reached, ηχ 1010 − 1011Pa, the ηχ-dependency becomes negative, Q ∼ η−1χ−1, consistent with the prediction for a Maxwell model (e.g. Efroimsky 2012). This indicates xice= 10% a progressive change from an Andrade-like behaviour to a x = 20% Maxwell behaviour with decreasing ηχ. For forcing periods ice between a few days and a few tens of days and viscosity between 16 18 10 xice= 50% 10 Pa.s and 10 Pa.s, ηχ typically ranges between 10 and 1012 Pa.s, indicating a predominance of the Andrade behaviour. A Maxwell regime can be reached only for very long forcing periods (> 100 days) or low viscosity values (≤ 1014 −1015 Pa.s).

We found that the Q factor variations as a function of ηχ can be qualitatively reproduced by the following relationship:

!−1 Q /2 2 1 Fig. 7. Computed tidal Love number k2 for ice-rich planets with mass min Q = + + B α , (5) ranging between 0.5 and 10 Earth’s mass and H20 mass fraction up to X QminX AiX i 50% relative to total mass. The lines represent the fit based on Eq. 3 and the polynomial coefficient provided in Table 3. where Qmin is the minimal value of Q and X = (ηχ)/(ηχ)Q=Qmin . We tried to derive the parameters Ai and Bi but we failed to find

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7. Applications to a selection of solid rocky and ice-rich exoplanets We applied the scaling laws derived here to a selection of exoplanets having a mass comprised between 0.5 and 10 M⊕, a composition either rocky or ice-rich and a low surface temperature in order to avoid surface melting (for ice or rock) (see Table 4). In order to derive a meaningful planet composition, we also chose planets for which both masses and radii were determined with a relatively good accuracy. For each planet, we determined their composition in term of iron content (for water-free rocky planets) or ice content (for ice-rich planets) Trappist-1f Trappist-1g compatible with the range of their observed mass and radius using the M-R relationship provided in Table 1. Then the Love number k2 was determined using the scaling laws provided in Table 3. The Q factor was estimated from the values computed for planet masses of 1, 5 and 10 M⊕ (see Figures 6 and 9) and then extrapolated to the corresponding planet mass. As it is shown in Figures 6 and 9, the Q factor strongly depends on the assumed viscosity. In Table 4, we provide the values estimated for a given viscosity proﬁle in order to highlight the sensitivity to planet composition. For rocky planets, the Q factor is calculated by considering the reference viscosity proﬁle (see section 2) using Eq. 4. For ice-rich planets, the Q factor is estimated for a viscosity of high-pressure ice of 1016 Pa.s, which can be considered as a reasonable lower limit (ηmin) (e.g. Durham et al. 1997). Table 4 and Figures 10 and 11 summarize the results obtained for six exoplanets: three rocky ones and three ice-rich ones. K2-240 c

Rocky planets – Kepler-36 b: This planet is consistent with an iron content

= 10% larger than the Earth’s, possibly up to 50%. However, due to x ice the uncertainties on the mass determination, an iron content

= 20% slightly lower than the Earth’s (down to -23.5%) is also x ice possible. A small fraction of water cannot also be excluded. = 50% But for simplicity, we consider only a rocky composition. x ice – LHS 1140 b: This planet is consistent with an iron content smaller than the Earth’s possibly down to -50%. Here again, due to the relatively large uncertainty on the mass determination, a larger iron content cannot be excluded. Alternatively to a low iron content, the mass and radius of this planet could be explained by a small fraction of water ice (up to 5-10% depending on the iron content). We consider here only the rocky case.

– Trappist-1 e: Among the three planets, this is the planet con- Fig. 9. Computed Q factor for ice-rich planets with mass of 1 (a), 5 (b) taining the larger iron content. Even if both radius and mass are determined with very good accuracy, the uncertainty on and 10 (c) M⊕, and three ice content (10, 20 and 50%), as a function of viscosity η and tidal frequency χ for the ice mantle. The vertical iron content remains rather large as the M-R relationship thick black segments indicate the expected values of Q for three planet is less sensitive to iron content in this mass range. As a candidates (Trappist-1 f, Trappist-1 g and K2-240 c) for a high-pressure consequence, an iron content slightly subterrestrial cannot ice viscosity of 1016Pa.s. The black arrows indicate the trend if larger be excluded. On the other hand, it is possible that the iron ice viscosity values are considered. content exceed 50% of the Earth’s value. However, as we did not perform any computation above this value, we cannot estimate with conﬁdence the upper limit in term of iron content. This translates in an indetermination of the upper limit for k2, indicated by a dotted line on Figure 10. a good ﬁt as a function of ice content comparable to what was The Love number, k2, is mostly sensitive to the iron content obtained for the rocky planet (see Figure 6). of the planet, and slightly to the planet mass for mass below

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Table 4. Title

Planet Reference Porb (days) M/M⊕ R/M⊕ x k2 Q

Rocky x = δFe Q(ηre f )

4.78 1.521 >0.5 0.488 − Kepler-36 b (Carter et al. 2012) 13.84 4.454.18 1.1491.451 0.164−0.235 0.4290.358 120 150 7.87 1.759 − 0.267 0.470 − LHS 1140 b (Ment et al. 2018) 24.73 6.986.09 1.7271.695 0.296<−0.5 0.3680.297 100 140 0.851 0.936 >0.5 >0.383 − Trappist-1 e (Grimm et al. 2018) 6.10 0.7720.697 0.910.883 0.457−0.116 0.3600.258 220 280

Ice-rich x = xice Q(ηmin)

5.3 a 2.1 >0.5 0.223 − K2-240 c (Díez Alonso et al. 2018) 20.52 4.64.3 1.81.7 0.4170.235 0.2090.163 3 20 1.014 1.075 0.163 0.230 − Trappist-1 f (Grimm et al. 2018) 9.21 0.9340.856 1.0461.016 0.11550.055 0.1850.156 50 100 1.246 1.179 0.258 0.188 − Trappist-1 g (Grimm et al. 2018) 12.35 1.1481.053 1.1481.115 0.1930.131 0.1620.143 20 50 a planet mass estimated from the M-R relationship of Weiss & Marcy (2014).

∼ 2 M⊕ . As the uncertainties on the iron content are rather sition comparable to the Earth). Its ice content may reach large, it results in a relatively large uncertainty on k2. As already values of 16%, and even larger values if iron-enrichment is illustrated in Figure 5, the Q factor is much less sensitive to considered in the rocky interior. the iron content and depends mostly on the forcing period and the assumed internal viscosity. As a consequence, assuming the – Trappist-1 g: This planet has an even lower surface temper- same reference viscosity structure, the planet LHS 1140 b has ature (199 K, (Gillon et al. 2017)) and ice content slightly the lowest Q factor among the three planets, as its orbital period higher than Trappist-1 g. Using our M-R relationship, we is the largest one. As already mentioned, the values of Q factor estimated the ice content to range between 13 and 26%. listed in Table 4 have been calculated assuming the reference viscosity structure and an α parameter ranging between 0.25. The viscosity may vary by a factor of 10-100 relative to the The computed Love number for ice-rich planets is about half reference value, which results in Q values possibly to 3-4 the values obtained for rocky planets at comparable masses. By larger. The values listed in Table 4 should be considered only as contrast, ice-rich planets are much dissipative that rocky ones, indicative. For any other viscosity and α values, the Q factor can with Q values possibly as low as 3 for instance in the case of be estimated using Eq. 4. K2-240 c. This, however, strongly depends on the assumed values for the viscosity of ice at high pressure, which is not well constrained. The Q values listed in Table 4 corresponds to the expected values for a viscosity of 1016 Pa.s, which should be Ice-rich planets considered as a lower estimate. For viscosity values ten times – K2-240 c: We chose this planet as a possible representative higher, Q can reach values exceeding 100 for Trappist-1 f and g, of ice-rich planets for mass of the order of 5 Earth mass. and thus become comparable to the Q factor predicted for rocky Among the catalog of detected exoplanets, we did not find exoplanets. The Q factor provided here is only indicative and any planet in this range of mass with surface temperature should be considered as a lower estimate. low enough to be compatible with cold icy surfaces. The surface temperature of K2-240 c is estimated to 389 K (Díez 8. Conclusion Alonso et al. 2018), which is the lowest temperature we found for this range of planet size. For this planet, only In the present study, we computed the tidal response of rocky the radius has been constrained from transit measurements. and ice-rich solid exoplanets for masses ranging between 0.5 The mass has been estimated only using M-R relationship, and 10 Earth masses. Accurate determination of the tidal param- following the law published by Weiss & Marcy (2014). eters of exoplanets (k2 and Q) taking into account their interior Radial velocity follow-up should be able to provide some structure and rheology is essential to better constrain their constraints on the mass in the near future. Based on the rotational and orbital history. We showed that the amplitude of published mass estimate, we concluded that this planet is an tidal response, characterized by the gravitational Love number, ice-rich planet with ice content approaching 50 %, possibly k2, depends mostly on the planet mass and the internal stratifi- even more. cation (mostly controlled by the bulk composition). We showed that, for a given planet composition, the tidal Love number, – Trappist-1 f: This planet has an estimated surface tempera- k2, is mostly controlled by self-gravity effect and increase as ture of 219 K, compatible with a cold icy surface. The mass a function of planet mass. For a given mass, it depends on the and radius constraints imply that the planet contain at least relative size of the iron core, and hence on the bulk iron fraction. 5% of water ice (assuming that the rocky part has a compo- For a given mass, the presence of outer ice layers reduces the

Article number, page 8 of 10 G. Tobie et al.: The tidal response of rocky and ice-rich exoplanets Trappist-1 e Trappist-1 Trappist-1f Trappist-1g LHS 1140 b Kepler-36 b Kepler-36 K2-240 c

Fig. 10. Predicted tidal love number, using Eq. 3, for a selection of three Fig. 11. Predicted tidal love number, using Eq. 3, for a selection of three rocky exoplanets (Trappist-1e, Kepler-36 b, LHS 1140 b) from their ice-rich exoplanets (Trappist-1 f, Trappist-1 g, K2-240 c) from their mass, radii and the iron content, δFe derived from the M-R relationship mass, radii and the ice content, xice derived from the M-R relationship provided in Table 1. provided in Table 1. amplitude of tidal response compared to rocky planets with no Q factor) for ice-rich planets compared to ice-free planets. This ice. The reduction is controlled by the ice fraction. For both is explained by the relatively low viscosity of ice, which is types of planet (rocky and ice-rich), we proposed relatively expected to be several order of magnitude below that of rocks. simple scaling laws to predict the Love number as a function of However, we should keep in mind that the effect of pressure is radius, planet mass and composition. not well constrained and that it might approach values similar to rock near the melting point (1018 − 1019 Pa.s). Viscosity is For the Q factor, we did not find any direct control by the known to be strongly dependent on temperature, and therefore planet mass, which contrasts with the prediction of Efroimsky both temperature and viscosity profile should be modeled in a (2012) assuming an homogeneous interior. However, as in consistent way. In the present work, we assumed, for simplicity, Efroimsky (2012), we observed a decrease of dissipation rate constant viscosity in each internal layer and did not take into with increasing mass (larger Q with large planets), but not for account coupling with the thermal evolution as it was done for the same reason. The reduction of dissipation rate is attributed instance in Behounkováˇ et al. (2011) dedicated to Earth-like here to the increase of mantle viscosity, which is expected to planets. Future works are required to determine the influence of increase with pressure and hence with planet size (Tackley thermal structure on the dissipation rate and the retroaction of et al. 2013), while Efroimsky (2012) attributed it as a result tidal dissipation on thermal evolution. of self-gravity effects. While prediction using homogeneous body formulation is a relatively good approximation for small The effect of external fluid envelop (water ocean, massive undifferentiated body, it does not describe correctly the tidal atmosphere) was also not taken into account. Preliminary tests response of large planets. The effect of density stratification, indicated that it significantly affects the tidal response of the the increase of elastic parameters and viscosity with increasing underlying solid layer and is sensitive to the size of the fluid pressure as well as the presence of a central liquid metallic core envelop. The presence of this external fluid envelop is expected cannot be taken into account with such a simplified formulation. to impact the dissipative properties of the planet by adding dissipative processes in the external envelop (e.g. Auclair-Desrotour Consistent with the results of Henning & Hurford (2014), et al. 2018) and by reducing the tidal response on the underlying we obtained a strong increase of dissipation rate (decrease of solid interiors and hence its dissipative properties. The coupling

Article number, page 9 of 10 A&A proofs: manuscript no. Tobie-AA_Tides_exoplanets between the external ﬂuid envelop and the planet interior will be addressed in a future study.

Acknowledgements. The research leading to these results has received funding from the European Research Council under the European Community?s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 259285).

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